Low-cost Accelerometers for Robotic Manipulator Perception

Morgan Quigley, Reuben Brewer, Sai P. Soundararaj, Vijay Pradeep, Quoc Le, and Andrew Y. Ng

Abstract— We present a series of experiments which explorethe use of consumer-grade accelerometers as joint position sen-sors for robotic manipulators. We show that 6- and 7-dof jointangle estimation is possible by using one 3-d accelerometer foreach pair of joints. We demonstrate two calibration approachesand experimental results using accelerometer-based control inboth position-control and torque-control regimes. We presenta manipulator design combining accelerometer-based sensingwith low-cost actuation, and conclude by demonstrating theutility of consumer-grade accelerometers even on high-precisionmanipulators.

I. INTRODUCTION

Many robotic applications demand manipulators with highdegrees of repeatibility, precision, and reliability. Production-line robots provide stereotypical examples of this domain.However, although high-precision manipulators excel at tasksfor which they can be pre-programmed, robotic manipulatorshave not become commonplace in the unstructured domainsof typical homes and workplaces.

Prior work has investigated the manipulation of objectswithout high-precision knowledge of its exact position, ori-entation, and shape [1]. Uncertainty must be resolved by therobot’s sensors, such as cameras, lasers, or contact sensors,among other modalities [2]. Furthermore, in some suchenvironments the task itself is not clearly defined, and mustbe inferred by the robot through human-robot interaction,scene inference, or other methods [3]. This body of researchdemonstrates that the effective manipulation error in unstruc-tured environments is no longer restricted to the mechanicalprecision of the manipulator; rather, it incorporates errorsfrom the entire robotic system. Sensor calibration, objectlocalization algorithms, and sensor-manipulator registration,among other sources of error, are cumulative and add upquickly. For many tasks envisioned for household servicerobots, millimeter-order repeatability errors are unlikely tocause drastic changes in the performance of the system, aserrors of that order are often overshadowed by larger errorsin the perception system, and thus the robotic system mustalready be robust to such perturbations.

In this paper, we present a sensing strategy which lever-ages the recent progress in low-cost 3-d MEMS accelerom-eters. Our approach mounts at least one 3-d accelerometerfor each pair of joints and infers the joint angles using an

M. Quigley, S. P. Soundararaj, Q. Le, and A. Y. Ng are with theComputer Science Department, Stanford University, Stanford, CA, USA{mquigley, saip, quocle, ang}@cs.stanford.edu

R. Brewer is with the Department of Mechanical Engineering, StanfordUniversity, Stanford, CA, USA rdbrewer@stanford.edu

V. Pradeep is with Willow Garage, Inc., Menlo Park, CA, USAvpradeep@willowgarage.com

Fig. 1. Two manipulators used to demonstrate the utility of accelerometer-based sensing: a Willow Garage PR2 Alpha and a prototype low-costmanipulator.

Extended Kalman Filter (EKF). Because of its low cost, thissystem of joint-position estimation can be used to augmentan existing robotic sensor suite, or to function as the primarysensor for a low-cost robotic manipulator.

II. RELATED WORK

Inertial sensing has been used extensively in recent yearsfor human motion capture. Several companies provide small,lightweight, and networkable inertial units which integrateaccelerometers, gyroscopes, and magnetometers, and can beeasily attached to human limbs and torsos. These units can beworn for use in film or video-game character modeling [4],virtual reality [5], [6], and activity recognition [7].

Accelerometers are used in virtually every aerial andunderwater robot as part of the attitude detemination system.However, accelerometers have found other uses in robotnavigation: for example, when strapped to legged robots, theycan be used to classify the surface, which can help in gaitselection and tuning [8].

Prior work using accelerometers in robotic manipulationincludes simulation results of configuration estimation [9],employing strapdown inertial units on heavy equipment [10],kinematic calibration [11], and the creation of a fault-detection system [12]. In [13], accelerometers were doubly-integrated using an Extended Kalman Filter (EKF) to esti-mate the configuration of a SCARA-type manipulator un-dergoing high-dynamic motions. The use of accelerometersin flexible-link robots was proposed in [14]. A human-robot system was proposed in [15] which incorporated theattachment of accelerometers and other sensors to a humantele-operator.

These prior studies have demonstrated that accelerometersoffer a compelling source of information. However, we

are not aware of published results of robotic manipulatorswhich emphasize the potential for cost-reduction and higherreliability by using accelerometers for state estimation andcontrol.

III. STATE ESTIMATION

Virtually all robotic manipulators use shaft encoders ofsome variety (optical, magnetic, resistive, etc.) to determinethe kinematic configuration of the manipulator. In this sec-tion, we depart from this convention and discuss a sensingscheme based solely on 3-d MEMS accelerometers.

In static conditions, a 3-axis accelerometer returns a 3-dvector pointed at the center of the earth. Since the length ofthis vector is fixed at 1 g, a 3-axis accelerometer only has twodegrees of freedom. Hence, one accelerometer is required forevery two rotary joints in a robotic manipulator. However, ifthe kinematics of the structure allows it, incorporating oneaccelerometer per rotary joint will increase the robustness ofthe calibration and the accuracy of the joint-state estimates.This effect will be discussed in the following sections.

A. Joint-angle estimation

In this section, we will describe how we produce a coher-ent estimate of the manipulator state using accelerometersattached to each link of the kinematic chain. This is a sensor-fusion problem: each accelerometer, by itself, can only deter-mine the direction of a downward-facing vector. However, byusing a priori knowledge of the kinematic constraints of themanipulator, it is possible to produce a unified state estimate.In our implementation, this is done by an Extended KalmanFilter (EKF). An augmented forward kinematics function isused to predict the accelerometer readings given the currentbelief state. The EKF algorithm then updates the beliefstate after observing the true measurement. The followingparagraphs will discuss this process in more detail.

First, we note that given a 3-d accelerometer reading andknowledge of the axis of rotation, it is possible to computethe joint angle in all cases except a singularity where theaxis of rotation is parallel to vertical. In an all-accelerometersensing scheme, this set of configurations must be avoided,and the severity of this limitation is dependent on the kine-matics of the manipulator and the required task: for example,accelerometer-only sensing will completely fail on SCARAmanipulators. However, for the anthropomorphic manipula-tors considered in this paper, vertical joint configurations arereadily avoidable. Furthermore, excursions through verticaljoint orientations could be gracefully handled by augmentingthe accelerometer measurements with magnetometers, back-EMF sensing, angular rate sensors, or conventional shaftencoders in the EKF framework.

Numerous strategies can be employed to infer the manipu-lator configuration from accelerometer readings. We chose touse an EKF due to its relatively simple implementation andfast on-line performance. A detailed discussion of the EKFalgorithm is beyond the scope of this paper and is presentedin many excellent texts [16].

To encourage smooth estimates of the joint position, wedefine the EKF state space to include the joint angles,velocities, and accelerations:

x =

θ

θ̇

θ̈

(1)

where θ represents the joint angles of the manipulator. Thestate (plant) update function implements numerical integra-tion and assumes constant acceleration:

f(x) =

θ + ∆tθ̇

θ̇ + ∆tθ̈

θ̈

(2)

The measurement function needs to predict sensor mea-surements z given a state x. As is often the case in EKFimplementations, the measurement function could be madearbitrarily complex to capture more and more propertiesof the system. However, we found that a measurementfunction which only predicts the acceleration due to gravitywas sufficient to handle the low-frequency regime of ourmanipulator. Adding additional terms to capture centripetalaccelerations and linear accelerations induced by joint-angleaccelerations did not change the performance, as will bediscussed in Section V.

To predict the measurement zi of a particular 3-d ac-celerometer αi given the state x,

zi = Rαii R

ii−1 (x) · · ·R1

0 (x)R0w

001

(3)

where Rαii represents the rotation from the accelerometer

frame to the frame attached to link i, the rotations Rii−1 (x)are the rotations between the link frames, and R0

w is therotation from the base of the manipulator to the world. Thefollowing paragraphs describe these rotations in more detail.Rαii is determined by how accerometer αi is physically

mounted on link αi. For our manipulator, it is a permutationmatrix followed by a rotation of a few degrees. Rαi

i is a staticparameter that can be estimated by the calibration processdescribed in the next section.Rii−1 (x) is determined by the axial orientation of link i

and link i − 1, as well as by the joint angle θi present inthe state x. The link twist can be statically estimated duringcalibration, but the joint angle must be recursively estimatedby the EKF.R0w is the rotation between the base of the manipulator

and the gravitational frame. If the manipulator is stationary,this rotation is constant and can be estimated by staticcalibration.1

1There are numerous situations where the R0w rotation is not constant and

needs to be estimated, but they lie in more extreme domains of robotics:manipulation on vehicles traveling on mountainous terrain, on spacecraft,or aboard ships in rough seas, for example. These situations are far beyondthe scope of this paper but could be readily addressed by estimating R0

wthrough an inertial unit or some other means.

Fig. 2. Left: an unpowered arm used to evaluate the calibration potential ofthe accelerometer-based sensing approach. Right: touching the end effectorto points on the calibration board.

Using the state-update and measurement functions, theEKF algorithm produces a recursive estimate of the mean andcovariance of the state as more timesteps and observationsare experienced.

B. Calibration

For a variety of reasons, the accelerometer axes of sensi-tivity are unlikely to be exactly aligned with the axes of thereference frames attached to manipulator links. Internal mis-alignments of MEMS accelerometer axes, misalignments in-curred when mounting accelerometer-equipped circuit boardsto the manipulator links, and manufacturing tolerances of thelinks themselves combine to result in rotations of severaldegrees between the axes of the accelerometers and theaxes of the kinematic frames. Fortunately, these mechanicalimperfections are static, and can be calibrated out.

To demonstrate calibration for low-cost manipulators usingaccelerometers as the primary sensors, we constructed theunactuated arm shown in Figure 2. This 4-dof mockup hasa roughly spherical shoulder and an elbow joint, with linklengths similar to the manipulator shown in Figure 1.

Our calibration scheme for the low-cost prototype isloosely derived from the checkerboard-based calibrationmethod widely used in computer vision [17]. A planarcalibration board was placed in the workspace of the ma-nipulator. The end effector of the manipulator was touchedto its corners, each time recording the corresponding ac-celerometer readings. The board was then moved and rotatedto a different position and orientation, and the process wasrepeated to collect a dataset of 20 different measurementscovering the manipulator workspace. This provides the scaleand skew constraints. This data was augmented by collectingmany accelerometer readings of manipulator configurationswhere the end effector was in contact with a large planarsurface such as a tabletop. This serves to provide a planarityconstraint.

Formally, the optimization problem can be written as:

arg minL,R

g1(α,L,R) + λ2g2(α,L,R) + λ3g3(R) (4)

where α are the accelerometer readings in the test set, Lrepesents the estimated link parameters, and R representsthe rotation matrices representing the misalignment of eachaccelerometer frame to its respective link frame.

The first term enforces the known scale of the calibrationboard.

g1(α,L,R) =∑i,j

||dij − d̂ij || (5)

where d̂ij is the distance between the estimated positions ofthe manipulator:

d̂ij = |FK(αi,L,R)− FK(αj ,L,R)| (6)

Here, FK means to run the forward-kinematics joint-angleestimation algorithm described in the previous section toproduce estimates of the end-effector positions using thelink parameters L. The subscripts i and j identify timesin the training set which were gathered from the sameposition and orientation of the calibration square, whichtherefore correspond to end-effector positions whose ground-truth distance dij is known.

The second term of the calibration optimization function,g2, corresponds to the planarity constraint of the end-effectorpositions gathered from the tabletop. Let P be the planefitted by taking the vectors corresponding to the top twosingular values of Y TY , where Y is the n x 3 matrix whoserows yi consist of the end-effector positions calculated bythe estimated calibration. The sum of the distances betweenthe end-effector positions and their projections onto the fittedplane provides an additional source of information about theseverity of the miscalibration. Then,

g2(α,L,R) =∑yi

||yi − projP (yi)|| (7)

The final term of the optimization function g3 encouragesthe misalignment rotation matricesRi to remain orthonormalduring the optimization:

g3(R) =∑i

||RiTRi − I3|| (8)

To calibrate accelerometer-based sensing on a high-precision robot, we used the Willow Garage PR2 Alphaplatform as a test bed. This robot has 7-dof manipulatorsequipped with accurate optical shaft encoders in each joint.The manipulator is well calibrated and the link parametersare known a priori. The calibration task is simplified, leavingonly the rotations Ri of the accelerometer mounting on thekinematic frame to be determined. We treat the joint anglesfrom the shaft encoders as ground truth and compare thesewith the angle estimates from the accelerometer readingsto calibrate out the rotations. The optimization problempresented below is similar to Equation 4

arg minR

∑i

||θi − θ̂i(α,Ri)||+ λ∑i

||RiTRi − I3|| (9)

Fig. 3. Hold-out test set error during the optimization convergence onprototype manipulator. The horizontal axis shows the iteration number, andthe vertical axis shows the mean of the miscalibrations. The numericaloptimization drives the average error from 11mm to 2mm.

Fig. 4. Hold-out test set error during the optimization convergence onWillow Garage PR2 Alpha. The horizontal axis shows the iteration number,and the vertical axis shows the mean error in joint angle estimates of theshoulder lift and the upper arm roll. The optimization drives the averageerror from 0.1 radians to 0.02 radians.

where θi is the joint angle position of the manipulator asgiven by the shaft encoders and θ̂i is the estimate based onaccelerometer readings. θ̂i is computed by solving for jointangles by considering inverse kinematics on pairs of links.

Our implementation requires approximately one hourof CPU time to reach convergence using the simplex-based optimization technique implemented by the MATLABfminsearch function. As is always the case with non-convex optimization, a good starting point is necessary toachieve a reasonable solution. We found that starting withthe parameters from the CAD model and assuming perfectsensor alignment resulted in reasonable solutions for our testdata.

To evaluate the performance of the calibration methodsquantitatively, we allowed the optimization algorithm to usetraining data containing several manipulator positions andcheckerboard orientations, and maintained a hold-out testset of several other orientations for evaluation purposes.The results demonstrate a significant calibration accuracyimprovement compared to inital guess. Figure 3 shows theconvergence of the algorithm from an initial end-effector

Fig. 5. The shoulder motors are evenly spaced around the center of motionto reduce their inertial contribution. The remote center of motion allows thelarge second and third motors to be direct-drive. The wrist differential isfriction-driven by rubber wheels under high compression. This mechanismprovides zero-backlash pitch and roll motion.

mean error of 11mm down to 2mm on our prototype ma-nipulator. Figure 4 shows convergence of the algorithm froma mean joint-estimate error of 0.1 radians (5.7 deg) in theshoulder lift to 0.02 radians (1.15 deg) on an instrumentedWillow Garage PR2 Alpha.

We note that 2mm mean end-effector localization erroris an order of magnitude worse than what is reported bymanufacturers of high-quality robotic manipulators sensedby shaft encoders. However, it is more accurate than thebest camera-manipulator calibration we have been able toachieve in several years of constructing and calibratingvision-guided systems with high-performance manipulators.We speculate that this level of calibration error would not bethe limiting factor of using low-cost localization approachesin a complete robotic system.

IV. A LOW-COST MANIPULATOR

To explore the feasibility of low-cost manipulation usinga purely accelerometer-based control scheme, we used a 6-dof manipulator that was constructed in our laboratory undera budgetary constraint of $1000 USD (Figures 1, 5, 6). Asthis manipulator incorporates several unconventional designfeatures, we will summarize its design in this section.

The shoulder operates in a spherical RRR configurationwith a remote center of motion. Although the shouldermotors are powerful, low-cost, and low-backlash (in thegearhead), they exhibit some cogging due to their ferrouscore, which affects the controllability of these joints. Unfor-tuantely, powerful low-cost motors suffer almost uniformlyfrom cogging, and this is a tradeoff that must be balancedin designing and controlling a low-cost arm.

A friction differential drive (Figure 5) provides the wristwith pitch and roll degrees of freedom. The differential isformed by belt-driven rubber wheels pressed firmly againsta thin aluminum veneer. Such wheels are low-cost, durable,and effective at transmitting high torques. The friction driveprovides zero-backlash and an inherent safety limit on thewrist: overloading results in slippage rather than damage tothe drivetrain.

The gripper has two independently-driven fingers, eachconsisting of a 4-bar linkage for parallel gripping (Figure 6).

Fig. 6. The elbow is driven via belt from a large motor in the shoulder. Thegripper is fabricated from lasercut polypropylene, using flexures to create adurable 4-bar mechanism. The thin belts are used only to turn potentiometersfor position feedback.

Fig. 7. Accelerometers were attached to the upper arm, forearm, andgripper of a PR2 Alpha robot.

Linkages are constructed from lasercut polypropylene flex-ures to avoid the mechanical complexity of using discreteparts. Flexures have the further advantages of zero backlash,extreme durability/fatigue resistance, and low-cost.

Linux-based software was written using the open-sourceRobot Operating System (ROS) platform [18]. Separateprograms were written for firmware communication (viathe Linux usb-serial driver), state estimation, joint-spacecontrol, teleoperation via joysticks, trajectory recording, andtrajectory playback/looping. The ROS framework handles thepeer-to-peer connections and provides logging, playback, andvisualization tools.

V. EXPERIMENTS

In this section, we present a series of experiments thatquantify the performance of accelerometer-based state esti-mation on two robots: the low-cost manipulator discussedin the previous section, and a Willow Garage PR2 Al-pha, a high-quality 7-dof manipulator. Accelerometers weredesigned into the motor control boards of the low-costmanipulator, which were rigidly attached to four links. ThePR2 Alpha was outfitted with strapdown accelerometers, asshown in Figure 7.

A. State Estimation

To quantify the performance of the calibration and joint-tracking systems, we affixed accelerometers to a WillowGarage PR2 Alpha robot (Figure 7). This 7-dof manipulatoris equipped with high-quality shaft encoders, which served as

Fig. 8. Tracking the forearm roll of the robot shown in Figure 7, showingthe encoder ground-truth (red) against the joint angle estimate from theaccelerometers (blue).

ground truth for this experiment. Its kinematic configurationincludes a vertical first joint (the “shoulder pan”), which wedid not attempt to estimate in these experiments, since thisjoint axis is always parallel to gravity.

Figure 8 demonstrates the tracking performance of onejoint (the forearm roll) as the manipulator was smoothlymoved through its workspace. The following table (in de-grees) shows the mean error throughout the trajectory, mea-sured as the difference between the shaft encoder readingsand the joint state estimates from the accelerometers.

Shoulder Lift Upper Arm Roll ElbowError (deg) 0.965 0.926 1.590

This experiment was done under near-ideal conditions: thePR2 Alpha arm uses spring balances for gravity compensa-tion and small coreless motors [19]. The mechanism is thusextremely smooth and well-behaved, avoiding any transientsor other anomalies as it travels the workspace.

B. Torque Control

Manipulators equipped with shaft encoders can often ig-nore the state-estimation problem when designing a controlscheme, as the quality of the state estimate is often inde-pendent of the state of the robot. In contrast, the quality ofthe state estimates inferred from the accelerometers can varywith the configuration and velocity of the manipulator. In thissection, we will discuss the ramifications of this property onthe behavior of the low-cost manipulator equipped only withaccelerometers.

For this experiment, we wrapped a proportional-integral(PI) controller around the state estimates produced by theEKF described in Section III. To reduce the efforts requiredof the PI controller, we employed active gravity compen-sation by using the Jacobian to compute the feed-forwardtorques necessary to float the manipulator links [20].

We found that the accelerometer-only sensing scheme canbreak down under high dynamic conditions. Specifically,linear accelerations induced by angular joint accelerations arenot modeled in the measurement prediction of Equation 3,

Fig. 9. Estimating the repeatability of the manipulator using optical-tracking data of the manipulator repeatedly cycling between two targetpositions. The plots show the difference between each stopping positionof the arm and its respective cluster centroid, in the horizontal plane (left)and a vertical plane (right). The mean deviation distance is 18.6mm. Axesare scaled in millimeters. 14 trials were run, all of which appear on thisplot.

and neither are the centripetal accelerations induced by theangular joint velocities. Interestingly, we found that addingthose terms did not improve the high-dynamic performanceof the manipulator, nor did adding a derivative term to the PIcontroller. We speculate that for such terms to be effective,the overall system must have a higher bandwidth, calibrationlevel, or measurement SNR.

As a result of the previous observation, the low-cost ma-nipulator could become unstable in high-dynamic situations.Furthermore, the measurement model of Equation 3 doesnot model contact forces; as a result, large contact forcesmay also induce instability. In either case, stability can beregained by quickly ramping down motor torques, whicheffectively slows the manipulator down until it re-enters astable region of the coupled sensing and control systems.

To quantify the repeatability of the low-cost manipulator,an active optical tracking device (Atracsys EasyTrack500)was used to obtain ground-truth position data of the endeffector. The tracking system has an accuracy of 0.2mm.The manipulator was placed in front of the optical tracker,with the optical beacon attached to the gripper tip.

The manipulator was commanded to cycle between twojoint configurations which required motion of at least 30degrees on all six joints of the manipulator (excluding thegripper fingers), resulting in end-effector travel of 34cm. Theoptical-tracking data was analyzed to extract the points wherethe manipulator had stopped, resulting in one cluster for eachof the target positions. Figure 9 shows an estimate of therepeatability of the manipulator, as measured by the deviationof each of these stopping positions from the mean of itscluster. The mean deviation was 18.6mm, and the maximumdeviation was 33.9mm.

C. Position Control

Our final experiment controlled the PR2 Alpha arm in adoorknob-grasping task (Figure 10). To avoid the instabilitieswitnessed in the low-cost manipulator experiment, we usedthe accelerometer to control the low-frequency trajectory ofthe manipulator, and used the PR2’s shaft encoders to sta-bilize the high-frequency behavior. Our accelerometer-basedcontroller sent relative joint angle commands to the PR2. Assuch, this controller could be used without shaft encoders

Fig. 10. In this experiment, accelerometer-based state estimation is usedto generate relative joint position commands, allowing a position-controlledrobot to repeatedly grasp a doorknob.

Fig. 11. Under relative position control, our control scheme is able toreach out and repeatedly grasp a doorknob. This plot shows a trace of onejoint as it undergoes relative joint angle commands from the accelerometer-based sensing scheme and simple setpoint-interpolation to derive small stepcommands.

on any manipulator with position-based actuators, such asstepper motor-based manipulators. Since the accelerometersfly on the actual links of the robot, their measurements arenot corrupted by link droop or backlash.

Our accelerometer-based, encoder-stabilized controllerguided the PR2 Alpha through a sequence of control pointsto repeatedly grasp a door handle in front of the robot.Trajectory tracking in this position-controlled scenario isshown in Figure 11.

VI. DISCUSSION

The work was motivated by the ever-increasing precisionof consumer-grade MEMS accelerometers, and the observa-tion that some anticipated future domains of robotics, such ashome service robots, possess many sources of error beyondmanipulator repeatability. In such scenarios, a reduction inthe repeatability of the manipulator may not drasticallyincrease the overall system error.

In general, the accelerometer-based sensing strategy re-moves complexity from the mechanism and increases thecomplexity of the calibration and control software. This strat-egy is motivated by the observation that complex software,unlike complex hardware, can be replicated at no cost.

Because adding accelerometers to an existing manipulatordesign is mechanically simple and incurs very little cost, thissensing approach is also suitable as a backup, or auxilliary,sensing strategy for manipulators equipped with shaft en-coders. The accelerometers could be used to bootstrap thepower-up sequence of manipulators: regardless of the con-figuration of the manipulator at power-up, an accelerometer-driven EKF will quickly converge to a reasonable estimate ofthe manipulator configuration. After an accelerometer-basedjoint configuration is estimated, the manipulator could use itsincremental encoders to safely and quickly reach the homingflags for each joint. Furthermore, accelerometers can provideinformation about impacts, drivetrain health (through spectralanalysis), and a continual “sanity check” for the incrementalencoders.

VII. CONCLUSION AND FUTURE WORK

We presented a low-cost sensing scheme based on 3-d MEMS accelerometers. The system produces coherent,absolute estimates of the kinematic configuration of a ma-nipulator. We performed experiments to quantify the perfor-mance of this scheme using both high-precision and low-costmanipulators. The accelerometer-based sensing algorithmcan be readily applied to any manipulator to augment itsstate estimation with very little hardware cost and trivialmechanical complexity.

We intend to continue developing the ideas presented inthis paper, exploring a variety of approaches to reduce thecost of actuation, sensing, and control of robotic manipu-lators in unstructured environments. To gain feedback andencourage replication, the mechanical and electrical designsfor the low-cost manipulator, as well as all firmware andsoftware used in the accelerometer-based sensing scheme,will be made freely available at http://openarms.stanford.edu

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